Rational functions from a free monoid into another are characterized by the finiteness of the index of some congruence naturally associated with the function. A sequential bimachine is constructed computing the function, which is completely canonical, and in some sense minimal. This generalizes the Nerode criterion and the minimal automaton of a rational language, and similar results for sequential functions.

Two negative results concerning the so-called acceptable sets of numbers are extended to the case or arbitrary context-free languages with the help of conventional analytic techniques.

This paper considers further how the decodability condition imposes restrictions on a set of code words. A generating function is defined that describes the composition of the code words. The relation between the generating function and a "full" set of code words is found. This relation shows that the sum of arbitrary probabilities associated with the words of a full set must be one. A full set of code words is one to which no code word can be added and still keep the set decodable. It is also shown that a full set is "completable." For a completable set of code words any string of symbols can be made into a sentence by adding a suitable prefix and a suffix.